R. Adamczak and M. Strzelecki, Modified log-Sobolev inequalities for convex functions on the real line, Studia Math, vol.230, issue.1, p.11, 2015.

C. Ané, S. Blachère, D. Chafaï, P. Fougères, I. Gentil et al., Sur les inégalités de Sobolev logarithmiques, Panoramas et Synthèses [Panoramas and Syntheses, vol.10, p.13, 2000.

F. Barthe, P. Cattiaux, and C. Roberto, Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry, Rev. Mat. Iberoam, vol.22, issue.3, p.7, 2006.
URL : https://hal.archives-ouvertes.fr/hal-00014138

F. Barthe and A. V. Kolesnikov, Mass transport and variants of the logarithmic Sobolev inequality, J. Geom. Anal, vol.18, issue.4, pp.921-979, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00634530

F. Barthe and C. Roberto, Dedicated to Professor Aleksander Pe?czy?ski on the occasion of his 70th birthday (Polish). MR 2052235 14 6, Modified logarithmic Sobolev inequalities on R, Potential Anal, vol.159, issue.3, p.14, 2003.

W. Beckner, A generalized Poincaré inequality for Gaussian measures, Proc. Amer. Math. Soc, vol.105, issue.2, p.3, 1989.

S. Bobkov and M. Ledoux, Poincaré's inequalities and Talagrand's concentration phenomenon for the exponential distribution, Probab. Theory Related Fields, vol.107, issue.3, p.12, 1997.

S. G. Bobkov and F. Götze, Exponential integrability and transportation cost related to logarithmic Sobolev inequalities, J. Funct. Anal, vol.163, issue.1, p.15, 1999.

P. Cattiaux, Hypercontractivity for perturbed diffusion semigroups, Ann. Fac. Sci. Toulouse Math, vol.2188585, issue.6, p.17, 2005.
URL : https://hal.archives-ouvertes.fr/hal-00011201

P. Cattiaux and A. Guillin, On quadratic transportation cost inequalities, J. Math. Pures Appl, vol.2257848, issue.9, p.16, 2006.

P. Cattiaux, L. Guillin, and . Wu, Some remarks on weighted logarithmic Sobolev inequality, vol.60, p.15, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00485124

I. Gentil, A. Guillin, and L. Miclo, Modified logarithmic Sobolev inequalities and transportation inequalities, Probab. Theory Related Fields, vol.133, issue.3, p.15, 2005.
URL : https://hal.archives-ouvertes.fr/hal-00001609

N. Gozlan, Poincaré inequalities and dimension free concentration of measure, MR 2946156, vol.46, p.20, 2010.

N. Gozlan, C. Roberto, and P. Samson, From dimension free concentration to the Poincaré inequality, Calc. Var. Partial Differential Equations, vol.52, issue.3-4, 2015.

R. Lata?a and K. Oleszkiewicz, Between Sobolev and Poincaré, Geometric aspects of functional analysis, Lecture Notes in Math, vol.1745, p.3, 2000.

M. Ledoux, The concentration of measure phenomenon, Mathematical Surveys and Monographs, vol.89, p.10, 2001.

L. Miclo, Quand est-ce que des bornes de Hardy permettent de calculer une constante de Poincaré exacte sur la droite?, Ann. Fac. Sci. Toulouse Math, issue.6, p.13, 2008.

R. T. Rockafellar, Convex analysis, 1970.

J. Shao, Modified logarithmic Sobolev inequalities and transportation cost inequalities in R n, Potential Anal, vol.31, issue.2, p.16, 2009.

M. Talagrand, A new isoperimetric inequality and the concentration of measure phenomenon, Geometric aspects of functional analysis (1989-90), Lecture Notes in Math, vol.1469, pp.94-124, 1991.

, MR 1361756 (97h:60016) 3, 17 24. , Transportation cost for Gaussian and other product measures, MR 1392331, vol.6, p.17, 1995.

F. Wang, MR 2127729 3, 7 26. , From super Poincaré to weighted log-Sobolev and entropy-cost inequalities, MR 2446080, vol.22, p.16, 2005.

, UPS -F-31062 Toulouse Cedex 9, France E-mail address: barthe@math.univ-toulouse, vol.5219, pp.2-097